Further Lemma REF  of REF  reveals that the last term in the above is a non-negative supermartingale. Therefore, we are able to use known methods for bounding self-normalised vector-valued martingales [1]}, [2]}. Let \(\omega \)  be a stopping time with respect to the filtration. Applying Lemma 9 of [2]} to the stopped martingale gives:
\(\mathbb {P}\left( {S_{\omega }}_{\bar{W}_{\omega }^{-1}} \ge R\sqrt{\log \left(\frac{\det \left(\bar{W}_{\omega }\right)}{\lambda ^{d}}\right) + 2\log \left(\frac{1}{\delta }\right)}\,\right)\le \delta \) 
